Limit of $r(k,k)^{1/k}$

Status partial high confidence

The existence of the limit $\lim_{k\to\infty} r(k,k)^{1/k}$ remains open. The lower bound gives $\liminf_{k\to\infty} r(k,k)^{1/k} \ge \sqrt{2}$ (Erd\u0151s 1947, sharpened by Spencer). In 2023, Campos, Griffiths, Morris, and Sahasrabudhe proved $r(k,k) \le (4-\varepsilon)^k$ for $\varepsilon = 2^{-7}$, the first exponential improvement over the 1935 Erd\u0151s\u2013Szekeres upper bound of $4^k$; a 2024 preprint reportedly pushed the upper bound further to approximately $3.799^k$. Neither result resolves whether the limit exists or determines its value.

Cited literature (1)

• partial An exponential improvement for diagonal Ramsey (2023)
  Campos, M., Griffiths, S., Morris, R., Sahasrabudhe, J. · arXiv preprint · arXiv:2303.09521

  Proves $r(k,k) \le (4 - 2^{-7})^k$, the first exponential improvement over the 1935 Erd\u0151s\u2013Szekeres bound of $4^k$; strictly shows $\limsup_{k\to\infty} r(k,k)^{1/k} < 4$, but does not settle the existence of the limit.

Reviewer notes. The related corpus record erdosproblems.com/77 tracks the same problem. The 2023 Campos--Griffiths--Morris--Sahasrabudhe result is a major partial advance: it shows $\limsup r(k,k)^{1/k} < 4$, narrowing the gap between the lower bound ($\sqrt{2} \approx 1.414$) and the new upper bound ($\approx 3.992$), but leaves the existence question open. Wikipedia (checked 2026-09-10) also mentions a 2024 preprint that reportedly improves the upper bound to approximately $3.799^k$, but that preprint URL could not be verified within the 8-call cap and is not cited. The arXiv listing for 2303.09521 shows a revision dated August 2025, possibly corresponding to a journal submission or acceptance, but no journal DOI was confirmed.

Auto-reviewed 2026-09-10 with claude-sonnet-4-6 (web search enabled).